Galois Representations, -series and a Conjecture of Artin

The Galois group is an object of central importance in number theory, and we can interpreted much of number theory as the study of this group. A good way to study a group is to study how it acts on various objects, that is, to study its representations.

Endow with the topology which has as a basis of open neighborhoods of the origin the subgroups , where  varies over finite Galois extensions of  . (Note: This is not the topology got by taking as a basis of open neighborhoods the collection of finite-index normal subgroups of .) Fix a positive integer  and let be the group of invertible matrices over  with the discrete topology.

Definition 9.5.1   A complex -dimensional representation of is a continuous homomorphism

For to be continuous means that if is the fixed field of , then is a finite Galois extension. We have a diagram

Remark 9.5.2   That is continuous implies that the image of is finite, but the converse is not true. Using Zorn's lemma, one can show that there are homomorphisms with image of order  that are not continuous, since they do not factor through the Galois group of any finite Galois extension.

Fix a Galois representation  and let be the fixed field of , so  factors through . For each prime that is not ramified in , there is an element that is well-defined up to conjugation by elements of . This means that is well-defined up to conjugation. Thus the characteristic polynomial of is a well-defined invariant of and . Let

be the polynomial obtain by reversing the order of the coefficients of . Following E. Artin [Art23,Art30], set

 (9.3)

We view as a function of a single complex variable . One can prove that is holomorphic on some right half plane, and extends to a meromorphic function on all .

Conjecture 9.5.3 (Artin)   The -function of any continuous representation

is an entire function on all , except possibly at .

This conjecture asserts that there is some way to analytically continue to the whole complex plane, except possibly at . (A standard fact from complex analysis is that this analytic continuation must be unique.) The simple pole at corresponds to the trivial representation (the Riemann zeta function), and if and is irreducible, then the conjecture is that extends to a holomorphic function on all .

The conjecture is known when . Assume for the rest of this paragraph that is odd, i.e., if is complex conjugation, then . When and the image of in is a solvable group, the conjecture is known, and is a deep theorem of Langlands and others (see [Lan80]), which played a crucial roll in Wiles's proof of Fermat's Last Theorem. When and the image of in is not solvable, the only possibility is that the projective image is isomorphic to the alternating group . Because  is the symmetry group of the icosahedron, these representations are called icosahedral. In this case, Joe Buhler's Harvard Ph.D. thesis [Buh78] gave the first example in which was shown to satisfy Conjecture 9.5.3. There is a book [Fre94], which proves Artin's conjecture for 7 icosahedral representation (none of which are twists of each other). Kevin Buzzard and the author proved the conjecture for 8 more examples [BS02]. Subsequently, Richard Taylor, Kevin Buzzard, Nick Shepherd-Barron, and Mark Dickinson proved the conjecture for an infinite class of icosahedral Galois representations (disjoint from the examples) [BDSBT01]. The general problem for is in fact now completely solved, due to recent work of Khare and Wintenberger [KW08] that proves Serre's conjecture.

William Stein 2012-09-24